3.3 Conservation of momentum in matrix
Similar to above, the stresses per unit length that act across matrix-melt interface are expressed from the perspective of the matrix as \(-\nabla\bullet H_{m}\). Momentum conservation in the matrix may then be simplified to the following:
\begin{equation} \left(5\right)\ 0=\ -\left(1-\phi\right)\nabla P_{m}+\ \nabla\phi\left(P_{m}-\ P_{f}\right)\ +\ \nabla\bullet\left[\left(1-\phi\right)\tau_{m}\right]+\ \rho_{m}\left(1-\phi\right)g-c\left(v_{m}-v_{f}\right),\nonumber \\ \end{equation}
where \(P_{m}\) is the pressure, \(\rho_{m}\) is the density, and\(\tau_{m}\) is the viscous stress tensor in the matrix phase. Bercovici et al. (2001) derived an expression for the interphase pressure difference using a materially invariant relation and a micromechanical model of pore collapse:
\begin{equation} {\left(6\right)\text{\ P}}_{m}-\ P_{f}=-\zeta\nabla\bullet v_{m},\nonumber \\ \end{equation}
where \(\ \zeta=\frac{\text{πξ}}{\phi},\ \)and \(\pi\) is a geometric constant, \(\xi\) is the effective matrix viscosity, equal to\(\frac{4}{3}\eta+K,\) and \(\eta\) and \(K\) are the shear and bulk viscosity of the matrix, respectively. Eq. (6) is consistent with the derivation of Bercovici et al. (2001) (see full derivation in the supplements), however, it deviates from the derivation of McKenzie (1984) (his eq. (C11)) and expressions often used in rock mechanics by a factor of 1/\(\phi\) (Connolly & Schmidt, 2022).
3.4.1 Boundary conditions – syringe experiments
To best approximate the phase separation experiments of Hoyos et al. (2022), we consider a compacting medium in a 1-D domain with a permeable upper boundary (at z = 0) and whose height, \(H\), decreases over time due to the upward migration of the lower boundary at a fixed boundary velocity, \(v_{H}\) (Fig. 2b ).
To simplify the boundary conditions for the momentum conservation equations, we introduce the segregation flux, \(S\), which is defined as:
\(\left(7\right)\ S=\ \phi\left(v_{f}-v_{m}\right)\).
At the bottom boundary, z = H(t), a no slip condition is assumed and\(v_{f}=v_{m}=\ v_{H}\) as the movement of matrix and fluid is coupled to the bottom boundary, which leads to \(S(z\ =\ H(t))=0.\)The combined mass conservation of fluid and matrix, upon substitution of\(S\), yields:
\(\left(8\right)\ \frac{\ \partial}{\partial z}\left[v_{m}+S\right]\)= 0,
which indicates that the sum \(v_{m}+S\) is constant and equals to\(v_{H}\) for 0 < z < H because of the bottom boundary condition. The upper boundary condition is impermeable to the solid material but not the interstitial liquid, so that\(S(z\ =\ 0)=\ v_{H}\).
3.4.2 Boundary conditions – centrifuge experiments
For application of the model to the centrifuge experiments, the boundary conditions are similar except that at the top of the column\(\frac{\ \partial}{\partial z}S=0\). This is because in this case, the matrix pressure, \(P_{m}\), and fluid pressure, \(P_{f}\), are equal at the top of the crystal column. Furthermore, \(v_{H}\) is no longer imposed and instead we need to track the location of the top of the sedimented (mushy) layer with time using the condition\(\frac{\ \partial}{\partial z}S=0\) at the top of the domain. This treatment allows for the development of a pure melt layer above the crystal column, which is an accurate description of the centrifuge experiments summarized in Connolly and Schmidt (2022). We assume that the particle settling is rapid with respect to the duration of each experiment. This is likely the case for all experiments other than ZOB9 (Connolly & Schmidt, 2022).